Diffusion of Innovation under Supply Constraints

In this paper we present a canonical setting that illustrate the need for explicitly modeling interactions between manufacturing and marketing/sales decisions in a firm. We consider a single firm that sells an innovative product with a given market potential and incurs lost sales when unable to supply the product due to capacity constraints. For such firms, we present a new, deterministic model of demand modified from the original model of Bass, that captures the effect of sales lost due to supply constraints on future demand. We use this model to plan production and sales in the firm with the objective of maximizing total sales during the lifetime of the product. We shoe that the trivial, myopic sales plan that sell the maximal possible amount at each time instant is not necessarily optimal. Using Pontrygagin’s maximum principle we show that the structure of the optimal sales plan is of the “build-up” type in which the firm does not sell at all for a period of time (even though it incurs lost sales during this period), and builds up enough inventory to never lose sales once it begins selling. We compute the optimal build-up period. Finally, we investigate the effects of changing model parameters, the benefits of initial inventory and delayed roll-put, the impact of the model on capacity planning and the effect of competition.